Formal Verification of Neural PDE Surrogates via SMT Compilation
A new research paper submitted to arXiv explores the formal verification of Fourier Neural Operators (FNOs), which are used to accelerate partial differential equation (PDE) simulations. While FNOs offer speed, they often lack guarantees regarding physical structure preservation. The authors demonstrate that with fixed weights and grids, the spectral convolution in an FNO acts as a linear map, allowing the forward pass to be represented exactly in Z3's linear real arithmetic. The study evaluates two encoding methods: an exact encoding that compiles spectral convolution into dense matrix multiplication for sound proofs, and a lighter frozen encoding that approximates the spectral path for faster but less rigorous checks. Experiments on small FNO surrogates for 1D advection-diffusion-reaction models show that the exact encoding successfully identifies positivity proofs and counterexamples, though it faces timeout issues with ReLU models. The frozen encoding scales better but sacrifices certification accuracy. These findings highlight the trade-off between soundness and scalability, offering insights into verifying production-scale neural operators.
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Formal Verification of Neural PDE Surrogates via SMT Compilation
A new research paper submitted to arXiv explores the formal verification of Fourier Neural Operators (FNOs), which are used to accelerate partial differential equation (PDE) simulations. While FNOs offer speed, they often lack guarantees regarding physical structure preservation. The authors demonstrate that with fixed weights and grids, the spectral convolution in an FNO acts as a linear map, allowing the forward pass to be represented exactly in Z3's linear real arithmetic. The study evaluates two encoding methods: an exact encoding that compiles spectral convolution into dense matrix multiplication for sound proofs, and a lighter frozen encoding that approximates the spectral path for faster but less rigorous checks. Experiments on small FNO surrogates for 1D advection-diffusion-reaction models show that the exact encoding successfully identifies positivity proofs and counterexamples, though it faces timeout issues with ReLU models. The frozen encoding scales better but sacrifices certification accuracy. These findings highlight the trade-off between soundness and scalability, offering insights into verifying production-scale neural operators.
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